Vol. 56, No. 2, 1975

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Subordination theorems for some classes of starlike fumctions

Roger W. Barnard and John Lawson Lewis

Vol. 56 (1975), No. 2, 333–366

Let Kr = {z : |z| < r},r > 0. For given α,0 < α < ,d,0 d < 1, and M,1 < M , let S(α,d,M) denote the class of univalent and normalized α starlike functions f in K1 with Kd f(K1) KM. The authors show the existence of a function F S(α,d,M) with the properties: (a) log F(z)∕z,z K1, is univalent, (b) if f S(α,d,M), then log f(z)lz,z K1, is subordinate to log F(z)lz,z K1. Letting α 0 they obtain a similar subordination result for normalized starlike univalent functions. They then point out that these subordination results solve and give uniqueness for a number of extremal problem in the above classes.

Mathematical Subject Classification
Primary: 30A32
Received: 4 December 1973
Published: 1 February 1975
Roger W. Barnard
John Lawson Lewis